Question: Given that $a$ is an odd multiple of $1183$, find the greatest common divisor of $2a^2+29a+65$ and $a+13$.
We can use the Euclidean Algorithm. \begin{align*}
&\text{gcd}\,(2a^2+29a+65,a+13)\\
&\qquad=\text{gcd}\,(2a^2+29a+65-(a+13)(2a+3),a+13)\\
&\qquad=\text{gcd}\,(2a^2+29a+65-(2a^2+29a+39),a+13)\\
&\qquad=\text{gcd}\,(26,a+13).
\end{align*}Since $a$ is an odd multiple of $1183$, which is an odd multiple of $13$, $a+13$ must be an even multiple of $13$. This means that $26$ is a divisor  of $a+13$, so the greatest common divisor is $\boxed{26}$.